Influence of Molecular Noise on the Growth of Single
Cells and Bacterial Populations
Mischa Schmidt1, Martin Creutziger1, Peter Lenz1,2*
1 Fachbereich Physik, Philipps-Universität Marburg, Marburg, Germany, 2 Zentrum für Synthetische Mikrobiologie, Philipps-Universität Marburg, Marburg, Germany
Abstract
During the last decades experimental studies have revealed that single cells of a growing bacterial population are
significantly exposed to molecular noise. Important sources for noise are low levels of metabolites and enzymes that cause
significant statistical variations in the outcome of biochemical reactions. In this way molecular noise affects biological
processes such as nutrient uptake, chemotactic tumbling behavior, or gene expression of genetically identical cells. These
processes give rise to significant cell-to-cell variations of many directly observable quantities such as protein levels, cell sizes
or individual doubling times. In this study we theoretically explore if there are evolutionary benefits of noise for a growing
population of bacteria. We analyze different situations where noise is either suppressed or where it affects single cell
behavior. We consider two specific examples that have been experimentally observed in wild-type Escherichia coli cells: (i)
the precision of division site placement (at which molecular noise is highly suppressed) and (ii) the occurrence of noiseinduced phenotypic variations in fluctuating environments. Surprisingly, our analysis reveals that in these specific situations
both regulatory schemes [i.e. suppression of noise in example (i) and allowance of noise in example (ii)] do not lead to an
increased growth rate of the population. Assuming that the observed regulatory schemes are indeed caused by the
presence of noise our findings indicate that the evolutionary benefits of noise are more subtle than a simple growth
advantage for a bacterial population in nutrient rich conditions.
Citation: Schmidt M, Creutziger M, Lenz P (2012) Influence of Molecular Noise on the Growth of Single Cells and Bacterial Populations. PLoS ONE 7(1): e29932.
doi:10.1371/journal.pone.0029932
Editor: Jörg Langowski, German Cancer Research Center, Germany
Received August 1, 2011; Accepted December 7, 2011; Published January 6, 2012
Copyright: ß 2012 Schmidt et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: This work was financed by the University of Marburg and the LOEWE programm of Hessen. The funders had no role in study design, data collection and
analysis, decision to publish, or preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.
* E-mail: peter.lenz@Physik.Uni-Marburg.DE
guarantee the survival of the population. In a similar way, a
population can survive nutritional stress conditions by having
some non-growing cells that sporulate [12].
Generally, the production of unneeded proteins leads to an
additional burden reducing the growth rate [13], [14]. However,
there are situations where this burden is compensated. For
example, for a population growing in a fluctuating environment
(with varying nutrients) synthesis of these additional proteins could
be useful for the individual cells. It is a priori not clear if it is better
to just produce the molecular machineries required to grow on the
currently present nutrients or to produce additional machineries
required for other (currently not present) nutrients. The first
strategy has the advantage that the protein burden is lower thus
leading to a higher growth rate for the current nutrient. However,
the drawback is that after a shift in the medium (or if the current
nutrient is running out) new molecular machinery required for
growth has to be produced leading to a lag phase. This machinery
possibly includes transporters, metabolic or catabolic enzymes and
additional ribosomes. In the second strategy, there is no lag phase
but the higher protein burden leads to a slower growth rate for all
different nutrients. In fact, both strategies can be favorable
depending on the timescale of the environmental fluctuations, the
duration of the lag phase and the growth rates supported by the
nutrients in the medium. As reviewed in [15], both strategies have
been observed for Escherichia coli. For example, an E. coli
population grown under glucose-limited conditions with a
Introduction
In recent years it has become clear that many biological
processes are intrinsically noisy leading to strong variations in
composition and properties of individual cells belonging to the
same population of genetically identical cells [1]. Important
examples include the delay times in uptake of nutrients [2],
variations in chemotactic tumbling behavior [3], entry into a
dormant state [4], [5], [6], sporulation and competence [7], [8]. In
many cases these variations are caused by noise at the
transcriptional or translational level [1], [9], [10]. Studies in
ecology and population genetics have shown that stochastic
variability in phenotype can have an advantageous effect on
populations growing in fluctuating environments. This effect is
known as bet-hedging [11]. Typically, in these systems members of
a population follow individual noise-induced strategies in preparation of environmental fluctuations. In particular, fluctuations in
the transcription process might result in synthesis of proteins that
are not required for growth in the given environment. Heterogeneous populations have been observed in different instances
especially if the environmental fluctuations pose a severe danger
for the population. An important example is that of entry into a
dormant state [4], [5], [6]. During dormancy cells cannot grow,
therefore, reducing the effective growth rate of the population.
However, in hostile environments, for example if the population is
exposed to antibiotics, the dormant cells do not die, and, thus,
PLoS ONE | www.plosone.org
1
January 2012 | Volume 7 | Issue 1 | e29932
Influence of Noise on Bacterial Populations
doubling time of 0:6h{1 keeps growing without a lag-phase when
transferred to a medium with excess of fructose, mannose, maltose,
and ribose. However, a lag phase occurs when the population is
transferred to a galactose or arabinose rich medium [16].
At the population level there can also be problems associated
with phenotypic variations. For example, a specific phenotypic
variation may increase the growth rate of individual cells but put
the whole population in danger if too many cells have this
phenotypic variation. In this case additional mechanisms are
required to stabilize the population against this variation. An
example is that of ‘cheater’ cells in yeast populations that do not
express FLO1 [17]. This protein is essential for flocculation that
guarantees the survival of populations in hostile environments that
contain high concentrations of antimicrobials or ethanol by
building up a protected space for the cells inside the floc. The
cheater cells have a smaller protein burden and, thus, a higher
growth rate. However, this phenotypic variation is not favorable
for the population as a whole since cheater cells do not contribute
to the flocculation process.
Phenotypic variations are also apparent in populations of E. coli
growing in nutrient rich conditions. Although the macroscopic
properties of bacterial populations are characterized by a few welldefined quantities such as growth rate, total cellular volume or
mass, DNA content, and the number of ribosomes [18] the
individual cells show large variations in birth size and even in
individual interdivision times [19], [20]. Interestingly, there are
also cellular processes that seem to be tightly regulated and that
are not affected by phenotypic variations. An important example is
cell division in E. coli. This process is precisely implemented and a
mother cell divides into two daughter cells that differ at most by 3–
10% in mass [19], [21], [22]. If this high precision is indeed a
consequence of a tight regulation, an evolutionary advantage in
cell division precision should be expected.
In this study we theoretically explore if there are evolutionary
benefits of phenotypic noise for a growing population of
genetically identical bacteria in non-hostile environments. In
particular, we analyze if specific experimentally observed regulatory responses to noise (i.e. suppression or allowance of noise) are
advantageous for a bacterial population growing in nutrient rich
conditions. We consider two specific examples: (i) Division site
placement in E. coli cells. In this process molecular noise is highly
suppressed. We theoretically analyze if a precise implementation
of the cell division process provides a growth advantage for a
bacterial population. (ii) The occurrence of noise-induced
phenotypic variations in fluctuating environments. Here, we
address the question if it is favorable for a bacterial population
to display phenotypic variations in a fluctuating environment, i.e.
to form a heterogeneous population in which both of the above
mentioned strategies are realized.
assume that it has a larger interdivision time (the time between
birth and subsequent division) than its larger sister.
In our model the growth of a bacterial population is simulated
by a sequence of cell division events. We start from one single
newborn cell and simulate growth and division of this cell and its
daughter cells. The individual doubling time TD (the time a cell
needs to double its mass) is set by the growth medium [18].
Because there are strong indications that cell division is coupled to
mass [23] we assumed that the mass of the mother cell at cell
division has a fixed value mD (that depends on the growth
medium). In the absence of noise the mother cell, thus, divides into
two daughter cells with birth mass mB ~mD =2. In this case, all
cells have the same interdivision time given by the prescribed
doubling time TD . However, if the division process is noisy, the
mother cell divides into two uneven daughter cells with mass mB
and mD {mB . In the simulations mB is given by a random number
drawn from a normal distribution centered around mD =2 with
standard deviation s (for details see ‘Exp. methods’). Moreover,
the two daughter cells of unequal size have different interdivision
times. We consider only the case of limited lack of precision to
make sure that both daughter cells contain a complete chromosome. To keep track of the division events in the population we
used the ‘‘time until division’’ (tud) that represents the time left
until an individual cell divides. At birth it is equal to the
interdivision time and in every time step of the simulations, the tud
of all cells is decreased by one (time unit). To calculate the tud
distribution from the mass distribution we used the fact that the
mass of an individual cell increases exponentially during its life
cycle [24], [25]. Correspondingly, the interdivision time of a cell
with birth math mB is given by TD ln(mD =mB )=ln 2. Thus, we
implicitly assume that (under the nutrient-rich conditions considered here) the smaller daughter cell only needs longer to reach the
doubling mass because of its smaller birth mass but does not have
a growth defect, as has been experimentally observed in [26].
Before using this model to investigate how cell division noise
affects the growth of the population, we first verified that in the
absence of noise our model produces an exponentially growing
population. To this end, we calculated an OD-plot by keeping
track of the total mass of the population as function of time. Figure
S1 clearly shows that the population indeed grows exponentially.
Similarly, we generated OD-plots for populations in the
presence of divisional noise. The doubling time of the population
Tpop is obtained from the slope of the OD plot. Surprisingly, we
found that Tpop does not depend on the standard deviation of birth
mass s that quantifies the divisional noise, see Figure S2. Thus,
noise in division site placement does not have any effect on the
growth rate of the population.
To find the origin of this interesting behavior, we developed an
analytical description of our numerical model. The key step is to
identify a formula for the tud distribution of the growing
population which can be obtained in a recursive fashion that
relates the tud distributions at time t and t{1
Results
Growth in non-fluctuating environments
We first analyzed the influence of single cell noise on
exponentially growing bacterial populations in a homogeneous
(non-changing) medium. As mentioned, there are many cellular
processes that could be affected by the presence of noise. Here, we
focused on its influence on the division process and we investigated
if the growth rate of a bacterial population depends on the
precision of cell division. In doing so we assume that an inexact
cell division event produces two sister cells of unequal length and,
consequently, of unequal mass. The smaller of the two daughter
cells has less mass and thus less ribosomes, less transporter proteins
and other molecular machinery needed to grow. We accordingly
PLoS ONE | www.plosone.org
n(x,t)~n(xz1,t{1)zn(0,t{1)P(x):
ð1Þ
Here, n(x,t) is the number of cells that have at time t a time until
division of x. In particular, n(0,t) is the number of cells that divide
at time t. P(x) is the probability distribution for the interdivision
times of the daughter cells that is obtained from the birth mass
distribution, for details see Eq. (15) in ‘Exp. methods’. It is
normalized to two, because every mother cell divides into two
daughter cells. The last equation states that a cell that has a tud of
x at time t either is a newborn cell or had a tud of xz1 at time
t{1. From Eq. (1) one can then show (as explained in detail in
2
January 2012 | Volume 7 | Issue 1 | e29932
Influence of Noise on Bacterial Populations
section 1 of Text S1) that the time evolution of the growing
population obeys the following equation
as follows from Eq. (2) (for details see section 2 of Text S1). Upon
using that individual cells increase their mass exponentially, i.e.
L
L
n(x,t)~ n(x,t)zn(0,t)P(x):
Lt
Lx
m(x)~mD 2{x=TD
ð5Þ
L
ln(2)
M(t)~
M(t):
Lt
TD
ð6Þ
ð2Þ
one then finds
Before analyzing Eq. (2) further we first tested if our method
reproduces the results of other approaches [27]. To do so we
calculated the steady state distribution ~n(x,t)~n(x,t)=N(t) of the
tud in the population. Here, N(t) is the total number of cells. As
shown in Fig. 1, in the absence of noise this distribution scales as
2x with the tud x (cyan line). In this case, the age y of a cell obeys
y~TD {x, and our results are in agreement with the classical
results on age distribution in growing bacterial populations [27].
As mentioned, n(0,t) is the number of dividing cells and this
quantity can be directly read off from the distribution of tud times.
Fig. 1 shows that this quantity is independent of the standard
deviation s.
Given this good agreement of our approach with classical results
we proceeded with the theoretical analysis of Eq. (2) to analyze
which ingredient of our model is responsible for the independence
of Tpop on s. To do so we analyzed the time dependence of the
total mass in the population
M ðtÞ~
?
ð
mðxÞnðx,tÞdx,
The last equation shows that the change in total mass does not
depend on the number of cells N(t) in the population. Thus, a
population of cells with an exponential mass increase always grows
exponentially with prescribed doubling time TD independently of
how the total mass is partitioned between the different cells. For
this reason the divisional noise does not affect the growth of the
population. This analysis clearly demonstrates that the experimental observation that E. coli cells increase their mass
exponentially is essential for our finding. For any non-exponential
individual mass increase [entering via Eq. (5) into Eq. (6)] the total
mass in the population depends on the number of cells leading to a
dependence on the precision of cell division.
Given the surprising result that single cell division noise has no
effect on the growth rate of a population we next asked whether noise
affects other population observables. Many cellular quantities such as
volume, mass, number of ribosomes proteins, or RNA content
change during the cell cycle, presumably in an exponential manner
[24], [25]. The distribution of all these quantities is easily calculated
from the tud distribution. For example, for the volume one has
ð3Þ
0
which changes with time as
Lt M ðtÞ~{
?
ð
V (x)~VD 2{x=TD ,
nðx,tÞLx mðxÞdx,
ð4Þ
ð7Þ
where VD is the volume at cell division. Thus, the volume distribution
is simply given by nðx,tÞV ðxÞ see Fig. S3. As one can see the tud
(Fig. 1) and the volume distribution look quite similar, they only have
a different scale on the x-axis. From this distribution one can easily
calculate the mean volume and the standard deviation, see Fig. S4. As
one can see these quantities have only a weak dependence on
divisional noise. This is, of course, a consequence of the rather small
differences in the tud distributions shown in Fig. 1 (for details see
section 3 of Text S1).
0
Growth in fluctuating environments
So far we have considered growth in homogeneous environments. In a next step we asked whether phenotypic variability
provides growth advantages in fluctuating environments. To
address this general question in a specific context we developed
a theoretical model that describes the growth of a bacterial
population in an environment with fluctuating supply of nutrients.
More specifically, we consider a situation where the nutrients in
the growth medium switch periodically (with period TS ). For
simplicity we consider the case of a periodic switching between two
limiting nutrients A and B.
There are mainly two strategies for how individual cells can
cope with these changing conditions. One strategy (in the
following referred to as strategy 1) is to synthesize only the
molecular machinery required to grow on the nutrient that is
currently present in the medium: if nutrient A is available only the
machinery for growth on A is produced and the machinery
required to grow on B is only produced if nutrient B is present.
Thus, if the nutrient switches, new molecular machinery has to be
synthesized. This requires an adaption time TA during which the
cells do not grow. After adapting to the presence of B and absence
Figure 1. Steady state distributions of time until division for
different noise levels. The steady state tud distributions n(x,t)=N(t)
(where n(x,t) is the number of cells that have at time t a tud of x and
N(t) is the total number of cells at time t) were calculated from our
model for different strength of divisional noise (quantified by the
standard deviation s of birth masses of the daughter cells). Data shown
are for s~0:2 (red), s~0:1 (green), s~0:05 (blue), s~0:02 (magenta)
and s~0 (cyan), where s is given in units of division mass. The
prescribed doubling time is TD ~60 min for all populations. For s~0
(cyan line) the distribution scales as 2x .
doi:10.1371/journal.pone.0029932.g001
PLoS ONE | www.plosone.org
3
January 2012 | Volume 7 | Issue 1 | e29932
Influence of Noise on Bacterial Populations
of A cells grow with doubling time T1 . For simplicity we assume
that the growth rate is identical for both growth conditions.
However, our results do not depend on this specific assumption.
A different strategy (in the following referred to as strategy 2) is
to produce all molecular machinery to grow on A and B
independent on whether A or B are currently present in the
medium. In this way, no adaption is required after a switch in
nutrients and the cells simply keep growing without a lag. Strategy
2 cells grow with doubling time T2 (again for both nutrients), and
because of the extra-burden of producing non-needed proteins one
expects T2 wT1 (i.e in the growth phase strategy 2 cells have a
smaller growth rate than strategy 1 cells), see Fig. 2.
Both strategies have been observed experimentally in different
situations, see Ref. [15] and references therein. We therefore first
analyzed if our model reproduces the experimental findings that,
depending on the values of the relevant parameters (adaption,
doubling and switching time), different strategies are advantageous.
To quantify this we determined the effective growth rate of
homogeneous populations (exclusively consisting either of cells that
use strategy 1 or strategy 2) in the fluctuating environment. For a
strategy 1 population the effective doubling time (calculated by
averaging over one period, see Fig. 2) is given by
T1eff ~
T1 TS
,
TS {TA
advantageous in the red regions of Fig. 3. The phase boundary
between these two regions (shown as grey line in Fig. 3) is given by
TA ~TS ð1{T1 =T2 Þ:
As shown in Fig. 3, the strategy 2 population grows faster for long
adaptation times and short switching times, i.e. when the ratio
TA =TS increases. More generally, both strategies can be advantageous depending on the adaptation time and the switching time.
Thus, both growth strategies are only advantageous in a limited
range of parameters (TA ,TS ). In a next step, we therefore asked if it
might be favorable for a population to display phenotypic variations
and form a population consisting of a noise-dependent mixture of
strategy 1 and strategy 2 cells. The idea behind this is that with such
a diversification a homogeneous population consisting of cells with,
say, strategy 1 might be able to extend the parameter range for
which it is advantageous by allowing some of its cells to convert to
strategy 2. To test this possibility, we need to investigate if there are
parameter values TA and TS for which such a diversified population
has the fastest growth.
To analyze this, we implemented two different models for
generating phenotypic variation as illustrated in Fig. 4. In both
models, the individual cells are either growing with strategy 1 or
with strategy 2. However, noise-induced fluctuations in gene
expression can lead to a change of the growth strategy of the
individual cells. For simplicity, we focus here on the case where
such a change occurs only in newborn cells right after birth. This
makes the following analysis easier but our conclusions do not
depend on this assumption. We consider two scenarios. In the first
scenario, described by model 1, only one of the two strategies is
stable meaning that one strategy is predominantly chosen by a
newborn cell. Here, the noise-induced fluctuations that lead to a
change in growth strategy at birth are short-lived and are diluted out
ð8Þ
while for a strategy 2 population one has
T eff
2 ~T2 :
ð9Þ
Fig. 3 shows the ratio T1eff =T2eff calculated from the last two
equations as function of adaptation time TA and switching time
TS . The blue region in Fig. 3 corresponds to the region in
parameter space where this ratio is larger than 1 and in which the
strategy 2 population grows faster, i.e. for these parameter values
strategy 2 is advantageous over strategy 1. In contrast, strategy 1 is
Figure 3. Optimal growth strategy for homogeneous populations. The ratio T1eff =T2eff of doubling times of homogeneous
populations growing with either strategy 1 or strategy 2 is shown for
varying adaptation times TA and switching times TS . The effective
doubling time T1eff of the population growing with strategy 1 is given
by Eq. (8), the effective doubling time T2eff of the population growing
with strategy 2 is given by Eq. (9). In the parameter range where
T1eff =T2eff is larger than 1 (region shown in blue) a strategy 2 population
is advantageous (i.e. the population with strategy 2 grows faster). In the
region where T1eff =T2eff is smaller then 1 (shown in red), strategy 1 is
advantageous. The grey line represents the phase boundary parameterized by Eq. (10). Data shown are for T1 ~60 min and T2 ~80 min.
Here, and in the following Figs. only the range TA vTS =2, is shown. For
higher values of TA , strategy 2 is trivially advantageous since strategy 1
cells stop growing as TA approaches TS .
doi:10.1371/journal.pone.0029932.g003
Figure 2. Growth curves of homogeneous populations in a
fluctuating environment. Time dependence of the number of cells
belonging to a homogeneous strategy 1 population (red curve) and a
homogeneous strategy 2 population (blue curve). Data shown are for a
switching time TS ~300 min, adaptation time TA ~180 min, and
doubling times T1 ~60 min and T2 ~80 min. In this example, strategy
2 is advantageous due to the high value of TA .
doi:10.1371/journal.pone.0029932.g002
PLoS ONE | www.plosone.org
ð10Þ
4
January 2012 | Volume 7 | Issue 1 | e29932
Influence of Noise on Bacterial Populations
second term accounts for cell division events of cells with strategy 1
that create newborn cells with strategy 1 and 2 with probability
12p and p, respectively. The third term accounts for the
corresponding cell division events of cells with strategy 2. Because
in this model there is no inheritance of growth strategy, the noiseinduced fluctuations are diluted out after one generation and the
probability p to have strategy 2 is independent of the strategy of
the mother cell. In contrast, in model 2 the strategy choice of a
newborn cell depends on the strategy of the mother. The
probability that mother and daughter cells have the same strategy
is 12p (i.e. p is the probability that a newborn cell has a different
strategy than its mother cell, see Fig. 4.). Correspondingly, for
model 2 one has
during one cell cycle. In the second scenario, described by model 2,
both strategies are stable and the system can switch between the two
strategies. In this case the noise-induced fluctuations are long-lived
and the choice of growth strategy can be passed from the mother to
the daughter cell. One possibility is to use an epigenetic inheritance
mechanism [28]. As a consequence, the switching event in a cell has
different effects on the growth strategy of its daughter cell in the two
models: for example, consider a mother cell that has switched from
strategy 1 to strategy 2. Then, in absence of any additional
fluctuations the mother and its daughter cells have different
strategies in model 1 but identical strategies in model 2.
Whenever a daughter cell has a different growth strategy than the
mother cell, it has to adapt to the external conditions. This also
requires an adaptation time, denoted by TAB . This adaptation process
is either caused by the synthesis of additional metabolic machinery
(for cells that switch from strategy 1 to strategy 2) or by the
adjustment to a higher growth rate (for cells that switch from strategy
2 to strategy 1). For simplicity, we focus in the following on the
special case where these two adaption times are equal: TAB ~TA . The
influence of this specific assumption on our results is discussed below.
To formulate these models analytically it is again, as for the case
of growth in homogeneous environments, convenient to use a
continuum description. Then, one has at time t, ni (x,t) cells using
growth strategy i with time until division x. At birth the
interdivision time of a newborn cell with strategy i is drawn from
a normal distribution Pi (x) with mean Ti and standard deviation
si for i~1 and 2. In the following, we use s1 ~s2 ~0:08 a typical
value of cell division noise [19], [21], [22]. However, our general
conclusions do not depend on this choice. By repeating the same
analysis that led to Eq. (2) for two different distribution functions
and by denoting the probability that a newborn cell has strategy 2
by p the time evolution equation for model 1 becomes
Ln1 (x,t) Ln1 (x,t)
~
z(1{p)n1 (0,t)P1 (x)zpn2 (0,t)P1 (x)
Lt
Lx
Ln2 (x,t) Ln2 (x,t)
~
zpn1 (0,t)P2 (x)z(1{p)n2 (0,t)P2 (x):
Lt
Lx
ð12Þ
In both models the total number of cells is given by
ð
N ðtÞ~ dxðn1 ðx,tÞzn2 ðx,tÞÞ,
ð13Þ
from which the growth curve of a diversified population can be
calculated. This allows the calculation of the doubling time Tdiv of
the diversified population (by fitting this growth curve with an
exponential).
To determine whether phenotypic variation can provide a
growth advantage in fluctuating environments we compared Tdiv
with the doubling times of the homogeneous populations by
calculating T1eff =Tdiv and T2eff =Tdiv . These quantities are shown as
function of adaptation and switching time for different diversification probabilities p in Figs. 5 and 6. From inspection of these plots it
becomes clear that a population displaying phenotypic variation can
only grow faster than one of the homogeneous population (regions
shown in blue), but never faster than both homogeneous
populations. For example in model 1 with diversification probability
p = 0.99, TA = 60 min and TS = 200 min the diversified population
grows faster than a homogeneous strategy 1 population (see Fig. 5A).
Ln1 (x,t) Ln1 (x,t)
~
z(1{p)n1 (0,t)P1 (x)z(1{p)n2 (0,t)P1 (x),
Lt
Lx
ð11Þ
Ln2 (x,t) Ln2 (x,t)
~
zpn1 (0,t)P2 (x)zpn2 (0,t)P2 (x):
Lt
Lx
In both equations, the first term on the right hand side describes
the decrease of the tud of every cell with increasing time. The
Figure 4. Division scheme in the two models for phenotypic diversification. A newborn cell randomly chooses a growth strategy. In model
1, strategy 2 is chosen with probability p, strategy 1 with probability 1{p independent of the strategy of the mother cell. In model 2 this scheme is
different, since here the probability of having strategy 1 or 2 depends on the strategy of the mother cell. Here, p denotes the probability that a
newborn cell has a different strategy than its mother cell.
doi:10.1371/journal.pone.0029932.g004
PLoS ONE | www.plosone.org
5
January 2012 | Volume 7 | Issue 1 | e29932
Influence of Noise on Bacterial Populations
However, for the same parameter values a homogeneous strategy 2
population grows faster (see Fig. 5B). Such a behavior is found for all
parameter values tested.
The conclusion that a homogeneous population is advantageous
over a heterogeneous population under all conditions tested is
surprising in light of earlier results [29],[30]. Several studies propose
that phenotypic variation can be advantageous in fluctuating
environments. To clarify which aspect of our model is responsible
for this different conclusion, we looked for advantages of
diversification under a variety of additional conditions.
As explained above, our model assumes periodic switching of
the external conditions. We asked if our findings change in a more
realistic scenario where TS is not constant but varies randomly. To
analyze this we implemented two different randomly switching
environments, where the switching time is drawn from a normal
distribution with mean TS and from an exponential distribution
with mean TS , respectively. As shown in Figure S5 and S6, both of
the ‘‘random switching environments’’ give rise to the same
population doubling times T1eff ,T2eff ,Tdiv as the environment with
periodic switching. This implies that the doubling times of all three
populations, T1eff ,T2eff ,Tdiv only depend on the average switching rate.
An important feature of our model is that cells have to adapt
when they have a different growth strategy than their mother cells.
Above, we assumed that this adaptation process takes the same
time as adaptation to an environmental switch, i.e. TAB ~TA . To
analyze the influence of this assumption, we systematically varied
the adaptation time TAB . For TAB wTA , we find, as above, that a
diversified population is always growing slower than one of the
homogeneous populations, since in this case the growth rate of the
diversified population is even further reduced. For decreasing
TAB vTA , the doubling time of the diversified population Tdiv
decreases continuously. As we are looking for situations where
diversification is advantageous, we can focus on the case TAB ~0.
Systematic analysis showed that even in this case phenotypic
Figure 5. Comparison of doubling times of homogeneous populations and populations showing phenotypic variation according to
model 1. The doubling time of a diversified population Tdiv and a homogenous population are compared for different adaptation times TA and
switching times TS . Single cell diversification is implemented according to model 1, see Fig. 4. Here the ratios T1eff =Tdiv (left column) and T2eff =Tdiv
(right column) are shown. If this ratio is larger than 1 (shown in blue), the heterogeneous population grows faster. Data shown are for diversification
probabilities p~0:99 (A and B), p~0:5 (C and D), and p~0:01 (E and F). In all cases the birth mass noise is 8% and data are only shown for TA vTS =2
as in Fig. 3.
doi:10.1371/journal.pone.0029932.g005
PLoS ONE | www.plosone.org
6
January 2012 | Volume 7 | Issue 1 | e29932
Influence of Noise on Bacterial Populations
Figure 6. Comparison of doubling times of homogeneous populations and populations showing phenotypic variation according to
model 2. The doubling times of a diversified population Tdiv and a homogenous population are compared for different adaptation times TA and
switching times TS . Single cell diversification is implemented according to model 2, see Fig. 4. Here the ratios T1eff =Tdiv (left column) and T2eff =Tdiv
(right column) are shown. If this ratio is larger than 1 (shown in blue), the heterogeneous population grows faster. Data shown are for diversification
probabilities p~0:99 (A and B), p~0:5 (C and D), and p~0:01 (E and F). In all cases the birth mass noise is 8% and data are only shown for TA vTS =2
as in Fig. 3.
doi:10.1371/journal.pone.0029932.g006
bacterial populations. It is well established that in hostile environments
phenotypic noise provides an advantage for a bacterial population [31].
For example, as shown experimentally an E. coli population that is
exposed to antibiotics survives by diversifying into a dormant
subpopulation [6]. Here, however, we study evolutionary benefits of
phenotypic noise for a growing population of genetically identical
bacteria in non-hostile and nutrient-rich environments.
Under these conditions we find that generally noise at the single
cell level has hardly any effect on the macroscopic properties of
bacterial populations and neither affects its growth nor its
composition (regarding e.g. mass and protein content). More
specifically, we analyzed if suppression of divisional noise or
allowance of noise in the transcriptional regulation of metabolic
machineries leads to an increased growth rate of a population.
Surprisingly, we found that in these scenarios the experimentally
observed regulatory scheme (i.e. suppression of noise in division
site placement and allowance of phenotypic variations for growth
diversification as described by model 1 does not provide a growth
advantage. However, in model 2 diversification can be advantageous, but only for large diversification probabilities pw0:5, see
Fig. 7. For example, for p~1, TA ~50 min and TS = 140 min one
has both T1eff =Tdiv w1 and T2eff =Tdiv w1 (i.e. this point in
parameter space is blue in both Figs. 7A and 7B). However, as
can be seen from Fig. 7C, the diversified population is only
advantageous in a small region of phase space indicating that some
fine-tuning of growth parameters (T1 ,T2 ) with environmental
parameters (TS ,TA ) is required. For decreasing p the parameter
range for which the diversified population is growing the fastest is
getting smaller until it vanishes at p~0:5.
Discussion
In this study we address the question if the absence or presence of
noise on the single cell level leads to evolutionary benefits for growing
PLoS ONE | www.plosone.org
7
January 2012 | Volume 7 | Issue 1 | e29932
Influence of Noise on Bacterial Populations
Figure 7. Population showing phenotypic variation according to model 2 growing without adaptation after a strategy switch. For
p~1 and TAB ~0, T1eff =Tdiv and T2eff =Tdiv are shown as function of switching time TS and adaptation time TA in Figs. A and B, respectively. Fig. C
shows min(T1eff ,T2eff )=Tdiv thus comparing the diversified population with the faster growing homogeneous population. In Fig. D the total number of
cells N(t) as function of time t is shown for the diversified population (black curve). The subpopulations growing with strategy 1 and 2 are shown in magenta
and cyan, respectively. Data shown are for TA ~50 min and TS ~140 min. In all cases the birth mass noise is 8% and data are only shown for TA v0:6TS .
doi:10.1371/journal.pone.0029932.g007
in fluctuating environments) does not lead to an increase in the
growth rate of the population. While it is well established that
molecular noise is essential for populations to cope with hostile
situations that require random decision making (such as sporulation
or entrance into competence or dormancy [4], [5], [6], [7], [8]) it is,
as demonstrated here, difficult to identify an evolutionary relevant
role of noise for bacterial growth in nutrient rich conditions. In fact,
in the examples considered in this work, noise becomes only
relevant for unrealistically high switching rates.
We found that divisional noise has no effect on the growth rate
of a population of cells with exponential mass increase (such as E.
coli). This finding is a direct consequence of the fact that (under the
nutrient-rich conditions considered here) the growth rate of the
individual cells is independent of cell size. Such a behavior has
been observed experimentally in [26]. Of course, it cannot be
excluded that under different conditions ‘non-linear’ effects
become important. For example, one could imagine that in hostile
environments smaller cells have a growth disadvantage making a
symmetric cell division favorable. However, to be able to
theoretically analyze such scenarios a systematic experimental
characterization (along the lines of [26]) of the growth behavior of
individual cells in hostile environments is required.
Given the fact that (in the scenarios considered) noise has no
effect on the growth rate it is surprising that the Min system
together with the nucleoid occlusion system, that determine the
site of cell division in E. coli, shows such high precision with cell
division occurring within 3%–10% (of total length) from mid-cell
[19], [21], [22]. Z-ring formation is even more precise [32].
Because precise cell division does not result in faster growth the
PLoS ONE | www.plosone.org
astonishing precision of cell division in E. coli is not the result of
optimization of growth rate in nutrient-rich conditions. As
mentioned, the precision could have some advantages under
conditions different from those considered here. Another possibility is that the Min proteins simply keep the FtsZ ring from forming
at the poles while the cell division site is determined by the action
of nucleoid occlusion [33], [34], [35], [36]. In this way physical
interactions associated with the position of the chromosomes
would be responsible for division site placement. Such mechanisms
where, e.g., the site of cell division is determined by the physical
properties of the membrane and the associated turgor pressure
have been intensively discussed in the literature [37], [38], [39].
Our results also indicate that for populations the presence of gene
expression noise does not necessarily lead to evolutionary advantages at
least not in the scenarios considered here, where such noise affects the
metabolic program of individual cells in a fluctuating environment. We
implemented two different models for generating phenotypic variation.
Our model 1 describes short-term fluctuations that last only for one
generation and are not inherited to the daughter cells. These
fluctuations may originate from uneven partitioning of ribosomes or
from improper sensing of the growth environment. Long-term
fluctuations that persist for longer than one cell cycle are taken into
account in our model 2 where the system can switch between the two
growth strategies. Typical examples are bistable switches that are
turned on or off by uneven partitioning of regulator proteins as
explained in [1]. The phenotypic state of these switches can be
inherited to the next generation [28].
Interestingly, for all (realistic) parameter values it is unfavorable
to form a noise-induced mixture of cells with different strategies.
8
January 2012 | Volume 7 | Issue 1 | e29932
Influence of Noise on Bacterial Populations
Our conclusion that diversification is (under realistic conditions)
not advantageous in fluctuating environments is very different
from those of other theoretical studies [29], [30]. However, we
believe that their findings are the consequence of an incomplete
comparison of all possible realizations of populations. In Ref. [29],
the authors consider a population with two subpopulations
growing at different growth rates. All cells can switch between
the two subpopulations. In this model an environmental change
simply leads to an exchange of growth rates of the subpopulations
that immediately continue growing with the new doubling time.
Thus, in contrast to our model the cells do not have to adapt to
environmental changes. Diversification is declared favorable, if a
population in which cells switch from the higher to the lower
growth rate grows faster than a population in which only switching
from low to high growth rate is allowed. However, this is only the
case if the population anticipates the fluctuations and the cells
switch into the state that will grow faster after the environment has
changed.
In Ref. [30], the authors consider two types of heterogeneous
populations. In one population, the cells choose the phenotype of
highest growth rate under the current conditions by sensing the
environment (responsive switching). In the other population, the
cells choose their phenotype randomly (stochastic switching). A
stochastic switching mechanism can lead to higher growth rate
than a responsive switching mechanism if the switching rates
between phenotypes follow the environmental changes. Thus, no
equivalent of our strategy 2 population (of cells that do not switch
phenotypes but grow with a burden) is considered.
In conclusion, these studies do not take into account that there
are two reference populations (in our terminology a homogeneous
strategy 1 and a homogeneous strategy 2 population) with which
the diversified population has to be compared. Most of the results
on advantageous effects of diversification are the consequence of
this incomplete comparison with the reference populations.
Furthermore, the models show that the strongest advantages
occur for populations that are able to anticipate the fluctuations in
the growth medium. As explained above, our model reproduces
these growth advantages for anticipating populations if the
adaptation process is neglected.
Our model is also applicable to more general situations than the
simple switching between two nutrients. For example, our results
and conclusions are not affected if a third nutrient C is present in
between the other nutrients A and B provided that the strategy 2
cells do not produce the machinery required to grow on C and
that both strategy 1 and strategy 2 cells need the same adaptation
time. Thus, our model not only describes the switching sequences
ABABAB but also ABCABC, ACBACB etc. and even scenarios
with non-periodic occurrences of C.
Things become more interesting if for optimal growth on C a
different strategy is required than for growth on A and B. But even
then our conclusion that a homogeneous population is always
favorable is not affected: both strategies lead to an effective
doubling time over the full switching time. The effective doubling
time of the mixture is then simply a linear superposition of the
effective doubling times of the 2 homogeneous populations. For
the case with two populations the optimal population then simply
implements the strategy with the lowest effective doubling time.
For more than 2 nutrients things are most interesting if the best
solution is to regulate the metabolic machinery of sugar B in the
presence of sugar A by one of the strategies and in the presence of
sugar C by the other strategy. But even in this case it is not
favorable to form a heterogeneous population because again the
doubling time of the mixture is a linear combination of those of the
homogeneous populations. Such a linear relation has its minimum
Because at least one of the homogeneous populations is growing
faster noise-induced diversification is unstable. These findings can
be understood as follows: for given growth conditions either
strategy 1 or strategy 2 is advantageous. Let’s assume that, strategy
1 is advantageous. Then, the fastest growing population consists
only of strategy 1 cells. For such a strategy 1 population
diversification leads to formation of a subpopulation of cells with
strategy 2. However, because strategy 2 cells grow (under the given
conditions) slower than strategy 1 cells this diversification just
implements the wrong strategy in some of cells leading to a
decrease in growth rate of the population. For a strategy 2
population, however, diversification leads to an increase in growth
rate since now some of the cells grow faster with strategy 1. In
particular, this population grows the faster the larger the fraction
of (diversified) strategy 1 cells is. Thus, in both cases the noiseinduced fluctuations drive the system towards a homogeneous
population with strategy 1. Similarly, for growth conditions that
favor strategy 2 the noise-induced fluctuations drive the system
towards a homogeneous strategy 2 population.
Thus, from an evolutionary point of view a diversified
population is not stable: for populations with the better strategy
every noise-induced fluctuation leading to a diversified population
makes the population less fit and the fluctuations die out. Also for
populations with the non-optimal strategy noise-induced fluctuations increase the growth rate. In this way the population is driven
towards a homogeneous population. This finding does not depend
on the details of the fluctuating environments and even holds for
random switching (where now TS is a random time) between the
two nutrients.
We found diversification to be favorable only for non-realistic
conditions. Namely, in our model 2 for phenotypic diversification
for a population that does not have to adapt to the change in
growth strategy at birth (TAB ~0) and that switches at high rates
pw0:5. That this corresponds to a rather artificial growth strategy
that also requires quite some fine-tuning of parameters can be
made clear by considering the case p~1. In this case all newborn
cells have a different strategy than their mother cell. Let’s consider
the case where a switching event occurs at t~0. Then, all strategy
1 cells stop growing (due to adaptation) while strategy 2 cells keep
growing and dividing. Because only strategy 2 cells divide and
p~1 the fraction of strategy 1 cells increases while that of strategy
2 cells decreases. Thus, for appropriately chosen TA the
population mainly consists of strategy 1 cells at t~TA . In this
way large parts of the population grow with the higher doubling
rate (T1 ){1 after the adaptation time is over. Thus, the
diversification strategy optimizes both, the growth in the lag
phase (by having a large fraction of strategy 2 cells for tvTA ) and
the growth after adaption (by having a large fraction of strategy 1
cells for twTA ). As growth proceeds the strategy 1 cells all divide
giving rise to a population that mainly consists of strategy 2 cells.
This leads to an oscillation of the composition of the population
(see Fig. 7D) that alternates between the two homogeneous
populations. Thus, the degree of diversification is not constant and
in contrast to the other scenarios there are no stable subpopulations. It is also clear that this strategy only works if T1 ,T2 ,TS and
TA are chosen properly. In fact, the population can only increase
its growth rate if it is able to anticipate the fluctuations in the
growth medium and adjust its growth parameters accordingly.
Such fine-tuning to environmental fluctuations can indeed lead to
growth advantages as was shown experimentally for an engineered
yeast strain in an accordingly chosen periodic environment [40].
However, since it involves parameter fine-tuning and unrealistic
high switching rates we don’t believe that this scenario has any
relevance for real biological systems.
PLoS ONE | www.plosone.org
9
January 2012 | Volume 7 | Issue 1 | e29932
Influence of Noise on Bacterial Populations
at the boundaries, i.e. either at f ~0 or f ~1 (where f is the
fraction of the population with one of the two strategies). Thus,
again the fastest growing population is a homogeneous one. The
sum of doubling times during the intervals AB and CB determines
whether it is strategy 1 or 2.
Given our main results that homogeneous populations always
grow faster than a mixed one, it becomes clear that the observed
variation in bacterial populations does not reflect an evolutionary
advantage under the conditions considered here. We can only
speculate here about its origin. For example, it could simply be of
physical origin, namely that the costs for a more precise regulatory
system exceed the benefit of being homogeneous. Or it might
provide a mechanism to keep phenotypic variations alive that
guarantee survival of the population under more severe or
irreversible changes in the environment.
Supporting Information
Text S1 Supporting text with derivation of equations.
(PDF)
Figure S1 OD-Plot of a growing population in the
absence of divisional noise. The OD plot was obtained by
calculating the total mass of the population M(t) as function of
time t. Mass is measured in units of the division mass mD , time t in
units of generation time TD . Population mass always doubles after
one generation showing that the population indeed grows with
prescribed doubling time TD .
(TIF)
Figure S2 Noise dependence of the population doubling
time Tpop . The doubling time of the population Tpop is obtained
by fitting the total mass of the population M(t) as function of time
t to an exponential. Tpop is shown in units of the prescribed
doubling time TD for different levels of birth mass noise quantified
bys.
(TIF)
Materials and Methods
The simulation of the growth of a bacterial population in a
homogeneous environment started from a single newborn cell at
time t~0 that divides at t~TD . To determine when cells divide
we keep track of the time until division x of every cell. A simulation
step represents one time step and in every time step x is reduced by
one for all present cells. At a given time all cells with x~0 divide
into two daughter cells. In the absence of noise, both daughter cells
have the same birth mass mB ~mD =2 (where mD is the division
mass of the mother cell) corresponding to an interdivision time of
TD . In the presence of noise, one daughter cell has birth mass mB
the other mD =2{mB , where mB is drawn from a normal
distribution with mean mD =2 and standard deviation s,
!
1
1 m{mD =2 2
:
P(m)~ pffiffiffiffiffiffi exp {
2
s
s 2p
Figure S3 Stationary volume distributions. The volume
histograms are shown for different strength of divisional noise:
s~0:2(red), s~0:1(green), s~0:05(blue), s~0:02(magenta) and
s~0(cyan).
(TIF)
and
Figure S4 Influence of noise on the mean volume V
the standard deviation sV of a population. A: The mean
volume (as given by Eq. 14 in Text S1) is measured in units of its
value for s~0. B: The standard deviation (calculated from Eq. 15
in Text S1) of the volume in units of its value for s~0.
(TIF)
ð14Þ
Figure S5 Effect of noisy switching on phenotypic
diversification described by model 1. T1eff =Tdiv is shown
as function of average switching time TS and adaptation time TA .
The diversification probability is p~0:99. In figures A and B, the
population growth was followed for 8 switching periods. In A,
switching times are drawn from a normal distribution, in B from
an exponential distribution. Figures C and D show averages over
100 runs shown in A and B, respectively. Birth mass noise is 8%.
We only consider environmental switches with TS w2TA .
(TIF)
Birth
mass is transformed
into the tud according to
m
x~{TD log2
. Correspondingly, in Eqs. (2), (11) and (12)
mD
one has then
P(x)~P(m(x))
m(x)ln2
,
TD
ð15Þ
where P(m) is given by Eq. (14) and m(x)~mD expð{lnð2Þx=TD Þ.
The growth of populations in fluctuating environments is
simulated by keeping track of the tud distributions n(x,t). For a
growing homogeneous population, the time evolution of the tud
distribution is given by Eq. (2). Strategy 1 and strategy 2 populations
behave like homogeneous populations except for the lag phase of
strategy 1 populations during which the tud distribution is kept
constant. The population doubling times are obtained by fitting the
total number of cells in the population
?
ð
N(t)~
n(x,t)dx
Figure S6 Effect of noisy switching on phenotypic
diversification described by model 2. T2eff =Tdiv is shown
as function of average switching time TS and adaptation time TA .
The diversification probability is p~0:01. In figures A and B, the
population growth was followed for 8 switching periods. In A,
switching times are drawn from a normal distribution, in B from
an exponential distribution. Figures C and D show averages over
100 runs shown in A and B, respectively. Birth mass noise is 8%.
We only consider environmental switches with TS w2TA .
(TIF)
ð16Þ
0
Acknowledgments
at time t with an exponential.
The diversified populations grow according to Eqs. (11) and
(12). Again, during the lag phase, the tud distribution of the
strategy 1 subpopulation n1 (x,t) is kept constant. The population
doubling time Tdiv is obtained by fitting the total number of cells,
Eq. (13), to an exponential.
For the simulations custom written C-programs were used.
PLoS ONE | www.plosone.org
We thank Lotte Søgaard-Andersen for many helpful discussions and a
critical reading of the manuscript.
Author Contributions
Analyzed the data: MS MC PL. Wrote the paper: MS MC PL. Designed
models and performed simulations: MS MC PL.
10
January 2012 | Volume 7 | Issue 1 | e29932
Influence of Noise on Bacterial Populations
References
20. Harvey RJ, Marr AG, Painter PR (1967) Kinetics of growth of individual cells of
Escherichia coli and Azotobacter agilis. J Bacteriol 93: 605–617.
21. Trueba FJ (1982) On the precision and accuracy achieved by Escherichia coli cells
at fission about their middle. Arch Microbiol 131: 55–59.
22. Guberman JM, Fay A, Dworkin J, Wingreen NS, Gitai Z (2008) PSICIC: noise
and asymmetry in bacterial division revealed by computational image analysis at
sub-pixel resolution. PLoS Comput Biol 4: e1000233.
23. Donachie WD, Begg KJ (1989) Cell length, nucleoid separation, and cell division
of rod-shaped and spherical cells of Escherichia coli. J Bacteriol 171: 4633–4639.
24. Ecker RE, Kokaisl G (1969) Synthesis of protein, ribonucleic acid, and
ribosomes by individual bacterial cells in balanced growth. J Bacteriol 98:
1219–1226.
25. Cooper S (1988) What is the bacterial growth law during the division cycle?
J Bacteriol 170: 5001–5005.
26. Schaechter M, Williamson JP, Hood JR, Koch AL (1962) Growth, cell and
nuclear divisions in some bacteria. Journal of general microbiology 29: 421–434.
27. Powell EO (1956) Growth rate and generation time of bacteria, with special
reference to continuous culture. J Gen Microbiol 15: 492–511.
28. Casadesus J, Low D (2006) Epigenetic gene regulation in the bacterial world.
Microbiology and Molecular Biology Reviews 70: 830–856.
29. Thattai M, van Oudenaarden A (2004) Stochastic gene expression in fluctuating
environments. Genetics 167: 523–530.
30. Kussell E, Leibler S (2005) Phenotypic diversity, population growth, and
information in fluctuating environments. Science 309: 2075–2078.
31. Davidson CJ, Surette MG (2008) Individuality in bacteria. Annual review of
genetics 42: 253–268.
32. Yu XC, Margolin W (1999) FtsZ ring clusters in min and partition mutants: role
of both the Min system and the nucleoid in regulating FtsZ ring localization.
Molecular Microbiology 32: 315–326.
33. Woldringh CL, Mulder E, Huls PG, Vischer N (1991) Toporegulation of
bacterial division according to the nucleoid occlusion model. Res Microbiol 142:
309–320.
34. Yu XC, Margolin W (1999) FtsZ ring clusters in min and partition mutants: role
of both the Min system and the nucleoid in regulating FtsZ ring localization. Mol
Microbiol 32: 315–326.
35. Margolin W (2001) Spatial regulation of cytokinesis in bacteria. Curr Opin
Microbiol 4: 647–652.
36. Errington J, Daniel RA, Scheffers DJ (2003) Cytokinesis in bacteria. Microbiol
Mol Biol Rev 67: 52–65.
37. Koch AL Bacterial growth and form: Springer.
38. Koch AL, Holtje JV (1995) A physical basis for the precise location of the
division site of rod-shaped bacteria: The Central Stress Model. MicrobiologySgm 141: 3171–3180.
39. Koch AL (2000) The bacterium’s way for safe enlargement and division. Applied
and environmental microbiology 66: 3657–3663.
40. Acar M, Mettetal JT, van Oudenaarden A (2008) Stochastic switching as a
survival strategy in fluctuating environments. Nature genetics 40: 471–475.
1. Smits WK, Kuipers OP, Veening JW (2006) Phenotypic variation in bacteria:
the role of feedback regulation. Nat Rev Microbiol 4: 259–271.
2. Megerle JA, Fritz G, Gerland U, Jung K, Radler JO (2008) Timing and
dynamics of single cell gene expression in the arabinose utilization system.
Biophys J 95: 2103–2115.
3. Spudich JL, Koshland DE (1976) Non-genetic individuality: chance in the single
cell. Nature 262: 467–471.
4. Shah D, Zhang Z, Khodursky A, Kaldalu N, Kurg K, et al. (2006) Persisters: a
distinct physiological state of E. coli. BMC Microbiol 6: 53.
5. Levin BR (2004) Microbiology. Noninherited resistance to antibiotics. Science
305: 1578–1579.
6. Balaban NQ, Merrin J, Chait R, Kowalik L, Leibler S (2004) Bacterial
persistence as a phenotypic switch. Science 305: 1622–1625.
7. Maughan H, Nicholson WL (2004) Stochastic processes influence stationaryphase decisions in Bacillus subtilis. J Bacteriol 186: 2212–2214.
8. Veening JW, Smits WK, Hamoen LW, Kuipers OP (2006) Single cell analysis of
gene expression patterns of competence development and initiation of
sporulation in Bacillus subtilis grown on chemically defined media. J Appl
Microbiol 101: 531–541.
9. Elowitz MB, Levine AJ, Siggia ED, Swain PS (2002) Stochastic gene expression
in a single cell. Science 297: 1183–1186.
10. Ozbudak EM, Thattai M, Kurtser I, Grossman AD, van Oudenaarden A (2002)
Regulation of noise in the expression of a single gene. Nat Genet 31: 69–73.
11. Philippi T, Seger J (1989) Hedging Ones Evolutionary Bets, Revisited. Trends in
Ecology & Evolution 4: 41–44.
12. Errington J (1993) Bacillus subtilis sporulation: regulation of gene expression and
control of morphogenesis. Microbiol Rev 57: 1–33.
13. Koch AL (1983) The protein burden of lac operon products. J Mol Evol 19:
455–462.
14. Dekel E, Alon U (2005) Optimality and evolutionary tuning of the expression
level of a protein. Nature 436: 588–592.
15. Kovarova-Kovar K, Egli T (1998) Growth kinetics of suspended microbial cells:
from single-substrate-controlled growth to mixed-substrate kinetics. Microbiol
Mol Biol Rev 62: 646–666.
16. Lendenmann U, Egli T (1995) Is Escherichia Coli Growing in Glucose-Limited
Chemostat Culture Able to Utilize Other Sugars without Lag. Microbiology
141: 71–78.
17. Smukalla S, Caldara M, Pochet N, Beauvais A, Guadagnini S, et al. (2008)
FLO1 is a variable green beard gene that drives biofilm-like cooperation in
budding yeast. Cell 135: 726–737.
18. Bremer H, Dennis PP (1996) Modulation of Chemical Composition and Other
Parameters of the Cell by Growth Rate In: Neidhart FC, ed. Escherichia coli and
Salmonella: Cellular and Molecular Biology 2nd ed. Washington DC: ASM Press.
pp 1553–1569.
19. Koppes LH, Woldringh CL, Nanninga N (1978) Size variations and correlation
of different cell cycle events in slow-growing Escherichia coli. J Bacteriol 134:
423–433.
PLoS ONE | www.plosone.org
11
January 2012 | Volume 7 | Issue 1 | e29932